44.4.3 problem 1 (c)
Internal
problem
ID
[7016]
Book
:
A
First
Course
in
Differential
Equations
with
Modeling
Applications
by
Dennis
G.
Zill.
12
ed.
Metric
version.
2024.
Cengage
learning.
Section
:
Chapter
2.
First
order
differential
equations.
Section
2.1
Solution
curves
without
a
solution.
Exercises
2.1
at
page
44
Problem
number
:
1
(c)
Date
solved
:
Tuesday, January 28, 2025 at 03:10:38 PM
CAS
classification
:
[_Riccati]
\begin{align*} y^{\prime }&=x^{2}-y^{2} \end{align*}
With initial conditions
\begin{align*} y \left (0\right )&=2 \end{align*}
✓ Solution by Maple
Time used: 0.816 (sec). Leaf size: 177
dsolve([diff(y(x),x)=x^2-y(x)^2,y(0) = 2],y(x), singsol=all)
\[
y = \left \{\begin {array}{cc} \frac {2 x \left (\pi \left (-\frac {\Gamma \left (\frac {3}{4}\right )^{2} \sqrt {2}}{2}+\pi \right ) \operatorname {BesselI}\left (-\frac {3}{4}, \frac {x^{2}}{2}\right )+\operatorname {BesselK}\left (\frac {3}{4}, \frac {x^{2}}{2}\right ) \Gamma \left (\frac {3}{4}\right )^{2}\right )}{\left (-\Gamma \left (\frac {3}{4}\right )^{2} \pi \sqrt {2}+2 \pi ^{2}\right ) \operatorname {BesselI}\left (\frac {1}{4}, \frac {x^{2}}{2}\right )-2 \operatorname {BesselK}\left (\frac {1}{4}, \frac {x^{2}}{2}\right ) \Gamma \left (\frac {3}{4}\right )^{2}} & x <0 \\ 2 & x =0 \\ \frac {x \left (\left (\sqrt {2}\, \pi \operatorname {BesselI}\left (-\frac {3}{4}, \frac {x^{2}}{2}\right )-2 \operatorname {BesselK}\left (\frac {3}{4}, \frac {x^{2}}{2}\right )\right ) \Gamma \left (\frac {3}{4}\right )^{2}+2 \pi ^{2} \operatorname {BesselI}\left (-\frac {3}{4}, \frac {x^{2}}{2}\right )\right )}{\sqrt {2}\, \pi \Gamma \left (\frac {3}{4}\right )^{2} \operatorname {BesselI}\left (\frac {1}{4}, \frac {x^{2}}{2}\right )+2 \pi ^{2} \operatorname {BesselI}\left (\frac {1}{4}, \frac {x^{2}}{2}\right )+2 \operatorname {BesselK}\left (\frac {1}{4}, \frac {x^{2}}{2}\right ) \Gamma \left (\frac {3}{4}\right )^{2}} & 0<x \end {array}\right .
\]
✓ Solution by Mathematica
Time used: 0.480 (sec). Leaf size: 146
DSolve[{D[y[x],x]==x^2-y[x]^2,{y[0]==2}},y[x],x,IncludeSingularSolutions -> True]
\[
y(x)\to \frac {\sqrt {2} \operatorname {Gamma}\left (\frac {3}{4}\right ) \left (x^2 \left (-\operatorname {BesselJ}\left (-\frac {5}{4},\frac {i x^2}{2}\right )\right )+x^2 \operatorname {BesselJ}\left (\frac {3}{4},\frac {i x^2}{2}\right )+i \operatorname {BesselJ}\left (-\frac {1}{4},\frac {i x^2}{2}\right )\right )-(2-2 i) x^2 \operatorname {Gamma}\left (\frac {1}{4}\right ) \operatorname {BesselJ}\left (-\frac {3}{4},\frac {i x^2}{2}\right )}{2 i \sqrt {2} x \operatorname {Gamma}\left (\frac {3}{4}\right ) \operatorname {BesselJ}\left (-\frac {1}{4},\frac {i x^2}{2}\right )+(2+2 i) x \operatorname {Gamma}\left (\frac {1}{4}\right ) \operatorname {BesselJ}\left (\frac {1}{4},\frac {i x^2}{2}\right )}
\]